Random matrix theory and chiral symmetry in QCD

Random matrix theory is a powerful way to describe universal correlations o f eigenvalues of complex systems. It also may serve as a schematic model fo r disorder in quantum systems. In this review, we discuss both types of app lications of chiral random matrix theory to the QCD partition function. We show that constraints imposed by chiral symmetry and its spontaneous breaki ng determine the structure of low-energy effective partition functions for the Dirac spectrum. We thus derive exact results for the low-lying eigenval ues of the QCD Dirac operator. We argue that the statistical properties of these eigenvalues are universal and can be described by a random matrix the ory with the global symmetries of the QCD partition function. The total num ber of such eigenvalues increases with the square root of the Euclidean fou r-volume. The spectral density for larger eigenvalues Glut still well below a typical hadronic mass scale) also follows from the same low-energy effec tive partition function. The validity of the random matrix approach has bee n confirmed by many lattice QCD simulations in a wide parameter range. Stim ulated by the success of the chiral random matrix theory in the description of universal properties of the Dirac eigenvalues, the random matrix model is extended to nonzero temperature and chemical potential. In this way we o btain qualitative results for the QCD phase diagram and the spectrum of the QCD Dirac operator. We discuss the nature of the quenched approximation an d analyze quenched Dirac spectra at nonzero baryon density in terms of an e ffective partition function. Relations with other fields are also discussed .